Hi, I'm trying to solve this excercise:
"Suppose that the complex number z has two m-th roots and that these are conjugated. is it true that z is real?"
I tried with this:
Consider [math]z = p (cos(\alpha)+isin(\alpha))[/math][math]\ = pe^{i(\alpha + 2k\pi)}[/math]Now let [math]w[/math] be the m-th root of [math]z[/math] ,so [math]w = p^{ \frac{1}{m}}e^{\frac{i(\alpha + 2k\pi)}{m}}[/math]Now the conjugated of [math]w[/math] is [math]p^{ \frac{1}{m}}e^{\frac{i(-\alpha + 2k\pi)}{m}}[/math] but we know tha w-conjugated is a m-th root also, so [math]w[/math] conjugated is also equal to [math]w = p^{ \frac{1}{m}}e^{\frac{i(\alpha + 2k\pi)}{m}}[/math].
We get
[math]p^{ \frac{1}{m}}e^{\frac{i(-\alpha + 2k\pi)}{m}} = p^{ \frac{1}{m}}e^{\frac{i(\alpha + 2k\pi)}{m}} \\ e^{\frac{i(-\alpha + 2k\pi)}{m}} = e^{\frac{i(\alpha + 2k\pi)}{m}} \\ \frac{i(-\alpha + 2k\pi)}{m} = \frac{i(\alpha + 2k\pi)}{m} \\ -\alpha + 2k\pi = \alpha + 2k \\ 2\alpha = 0 \\ \alpha = 0[/math]
Now if [math]\alpha = 0[/math] [math]z = p[/math], so [math]z\ is \ real[/math]
What do you think?
"Suppose that the complex number z has two m-th roots and that these are conjugated. is it true that z is real?"
I tried with this:
Consider [math]z = p (cos(\alpha)+isin(\alpha))[/math][math]\ = pe^{i(\alpha + 2k\pi)}[/math]Now let [math]w[/math] be the m-th root of [math]z[/math] ,so [math]w = p^{ \frac{1}{m}}e^{\frac{i(\alpha + 2k\pi)}{m}}[/math]Now the conjugated of [math]w[/math] is [math]p^{ \frac{1}{m}}e^{\frac{i(-\alpha + 2k\pi)}{m}}[/math] but we know tha w-conjugated is a m-th root also, so [math]w[/math] conjugated is also equal to [math]w = p^{ \frac{1}{m}}e^{\frac{i(\alpha + 2k\pi)}{m}}[/math].
We get
[math]p^{ \frac{1}{m}}e^{\frac{i(-\alpha + 2k\pi)}{m}} = p^{ \frac{1}{m}}e^{\frac{i(\alpha + 2k\pi)}{m}} \\ e^{\frac{i(-\alpha + 2k\pi)}{m}} = e^{\frac{i(\alpha + 2k\pi)}{m}} \\ \frac{i(-\alpha + 2k\pi)}{m} = \frac{i(\alpha + 2k\pi)}{m} \\ -\alpha + 2k\pi = \alpha + 2k \\ 2\alpha = 0 \\ \alpha = 0[/math]
Now if [math]\alpha = 0[/math] [math]z = p[/math], so [math]z\ is \ real[/math]
What do you think?